89 research outputs found
New holomorphically closed subalgebras of -algebras of hyperbolic groups
We construct dense, unconditional subalgebras of the reduced group
-algebra of a word-hyperbolic group, which are closed under holomorphic
functional calculus and possess many bounded traces. Applications to the cyclic
cohomology of group -algebras and to delocalized -invariants of
negatively curved manifolds are given
Property (RD) for Hecke pairs
As the first step towards developing noncommutative geometry over Hecke
C*-algebras, we study property (RD) (Rapid Decay) for Hecke pairs. When the
subgroup H in a Hecke pair (G,H) is finite, we show that the Hecke pair (G,H)
has (RD) if and only if G has (RD). This provides us with a family of examples
of Hecke pairs with property (RD). We also adapt Paul Jolissant's works in 1989
to the setting of Hecke C*-algebras and show that when a Hecke pair (G,H) has
property (RD), the algebra of rapidly decreasing functions on the set of double
cosets is closed under holomorphic functional calculus of the associated
(reduced) Hecke C*-algebra. Hence they have the same K_0-groups.Comment: A short note added explaining other methods to prove that the
subalgebra of rapidly decreasing functions is smooth. This is the final
version as published. The published version is available at: springer.co
Uniformizing the Stacks of Abelian Sheaves
Elliptic sheaves (which are related to Drinfeld modules) were introduced by
Drinfeld and further studied by Laumon--Rapoport--Stuhler and others. They can
be viewed as function field analogues of elliptic curves and hence are objects
"of dimension 1". Their higher dimensional generalisations are called abelian
sheaves. In the analogy between function fields and number fields, abelian
sheaves are counterparts of abelian varieties. In this article we study the
moduli spaces of abelian sheaves and prove that they are algebraic stacks. We
further transfer results of Cerednik--Drinfeld and Rapoport--Zink on the
uniformization of Shimura varieties to the setting of abelian sheaves. Actually
the analogy of the Cerednik--Drinfeld uniformization is nothing but the
uniformization of the moduli schemes of Drinfeld modules by the Drinfeld upper
half space. Our results generalise this uniformization. The proof closely
follows the ideas of Rapoport--Zink. In particular, analogies of -divisible
groups play an important role. As a crucial intermediate step we prove that in
a family of abelian sheaves with good reduction at infinity, the set of points
where the abelian sheaf is uniformizable in the sense of Anderson, is formally
closed.Comment: Final version, appears in "Number Fields and Function Fields - Two
Parallel Worlds", Papers from the 4th Conference held on Texel Island, April
2004, edited by G. van der Geer, B. Moonen, R. Schoo
Polarization and wavelength agnostic nanophotonic beam splitter
High-performance optical beam splitters are of fundamental importance for the
development of advanced silicon photonics integrated circuits. However, due to
the high refractive index contrast of the silicon-on-insulator platform, state
of the art Si splitters are hampered by trade-offs in bandwidth, polarization
dependence and sensitivity to fabrication errors. Here, we present a new
strategy that exploits modal engineering in slotted waveguides to overcome
these limitations, enabling ultra-wideband polarization-insensitive optical
power splitters, with relaxed fabrication tolerances. The proposed splitter
relies on a single-mode slot waveguide which is transformed into two strip
waveguides by a symmetric taper, yielding equal power splitting. Based on this
concept, we experimentally demonstrate -30.5 dB polarization-independent
transmission in an unprecedented 390 nm bandwidth (1260 - 1650 nm), even in the
presence of waveguide width deviations as large as 25 nm
Quantum Symmetries and Strong Haagerup Inequalities
In this paper, we consider families of operators in
a tracial C-probability space , whose joint
-distribution is invariant under free complexification and the action of
the hyperoctahedral quantum groups . We prove a strong
form of Haagerup's inequality for the non-self-adjoint operator algebra
generated by , which generalizes the
strong Haagerup inequalities for -free R-diagonal families obtained by
Kemp-Speicher \cite{KeSp}. As an application of our result, we show that
always has the metric approximation property (MAP). We also apply
our techniques to study the reduced C-algebra of the free unitary
quantum group . We show that the non-self-adjoint subalgebra generated by the matrix elements of the fundamental corepresentation of
has the MAP. Additionally, we prove a strong Haagerup inequality for
, which improves on the estimates given by Vergnioux's property
RD \cite{Ve}
Cycles in the chamber homology of GL(3)
Let F be a nonarchimedean local field and let GL(N) = GL(N,F). We prove the
existence of parahoric types for GL(N). We construct representative cycles in
all the homology classes of the chamber homology of GL(3).Comment: 45 pages. v3: minor correction
Nonlinear spectral calculus and super-expanders
Nonlinear spectral gaps with respect to uniformly convex normed spaces are
shown to satisfy a spectral calculus inequality that establishes their decay
along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to
behave sub-multiplicatively under zigzag products. These results yield a
combinatorial construction of super-expanders, i.e., a sequence of 3-regular
graphs that does not admit a coarse embedding into any uniformly convex normed
space.Comment: Typos fixed based on referee comments. Some of the results of this
paper were announced in arXiv:0910.2041. The corresponding parts of
arXiv:0910.2041 are subsumed by the current pape
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